Simple linear algebra question

Question

Let's say $A$ is an orthogonal $2\times2$ matrix over $\bf C$ and not diagonalizable over $\bf C$. Why then the determinant of $A$ must be $1$?

I guess I'm missing something easy...

Answer 1

Orthogonal matrices are normal, and all normal matrices are diagonalizable over the complex numbers.

Answer 2

This is false, e.g. $$\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$$

Answer 3

Edit: It is well known that every real orthogonal matrix is diagonalizable over $\mathbb{C}$. Actually, every $2\times 2$ complex orthogonal matrix is also diagonalizable over $\mathbb{C}$. See theorem 1.2.3 of this thesis, for instance. So, your assertion can be viewed as a vacuous truth.